GCSE Maths: The language of algebra (Higher)
Part of the GCSE Maths algebra basics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
The language of algebra
Algebra is a way of writing about numbers when we do not know the number yet, or when the number can change. Instead of a box or a question mark we use a letter. A letter that stands for a number is called a variable (or an unknown). For example, if a pack holds p pencils, then 5 packs hold 5 × p pencils, whatever p is.
Mathematicians are lazy writers, so algebra has short-cuts. Learn these rules first. You will see them on every page of this book.
| In words | Written in algebra | Why |
|---|---|---|
| a times b | ab | the × sign is left out |
| y + y + y | 3y | and 3y also means 3 × y |
| a × a | a2 | the little 2 is the index (or power) |
| a × a × a | a3 | three copies multiplied |
| a × a × b | a2b | the 2 belongs to a only |
| a ÷ b | ab | the division sign becomes a fraction line |
| half of x | x2 (not 0.5x) | in algebra we write a fraction rather than a decimal |
| 2 times (a + b) | 2(a + b) | a number next to a bracket means multiply |
The vocabulary of algebra
Algebra has its own words. Examination questions use them without explaining them, so make sure you can say what each one means.
| Word | Meaning | Example |
|---|---|---|
| Term | one part of an expression, joined to the others by + or − | 4x2, −7x and 5 are the terms of 4x2 − 7x + 5 |
| Coefficient | the number multiplying the letters in a term | the coefficient of 4x2 is 4; the coefficient of x is 1 |
| Constant | a term that is just a number | the 5 in 4x2 − 7x + 5 |
| Factor | something that is multiplied; it divides exactly into the term | 2 and x are factors of 2x; (x + 1) is a factor of 3(x + 1) |
| Expression | letters, numbers and operations, with no equals sign | 3x + 2 |
| Equation | two expressions joined by =, true for particular values of the letter | 3x + 2 = 11 (true only when x = 3) |
| Formula | a rule linking quantities, with several letters, that you use by substituting | A = lw (area of a rectangle) |
| Identity | an equality that is true for every value of the letter; written with ≡ | 2(x + 3) ≡ 2x + 6 |
| Inequality | two expressions compared with <, >, ≤ or ≥ | x + 1 > 4 |
Key ideas
- A letter stands for a number. 3y means 3 × y, and ab means a × b.
- a2 = a × a and a3 = a × a × a. The index only belongs to the letter it sits on.
- a ÷ b is written ab. Write fractional coefficients as fractions (x2), not decimals (0.5x).
- An expression has no equals sign. An equation has one and can be solved. A formula links quantities. An identity is true for every value.
- The sign in front of a term belongs to that term.
Method. Use the short-cuts: leave out × signs, number first, repeated letters become powers, divide becomes a fraction.
- Recall the rules. Every short-cut is just a way of writing the same thing in less space.ab means a × b; 3y means 3 × y; a2 means a × a
- Do (a) and (b): just remove the × sign and put the number first. The number goes at the front by convention.p × q = pq
4 × m = 4m - Do (c): count the repeated letters. Three copies of n multiplied together is n cubed, written with an index 3.n × n × n = n3
- Do (d): the number first, then each letter, grouping the repeated one. There is one a and two bs.5 × a × b × b = 5ab2
- Do (e): a division becomes a fraction. The fraction bar means divide.k ÷ 5 = k5
Check. Put a = 2 and b = 3 into (d): 5 × 2 × 3 × 3 = 90 and 5ab2 = 5 × 2 × 32 = 5 × 2 × 9 = 90. ✓
Watch out. In 5ab2 only the b is squared, not the 5 and not the a.
Key words. A term is one part of an expression. The coefficient is the number multiplying the letters. A constant is a term with no letter.
Method. Split the expression into terms at each + or − sign, and keep each sign with the term after it.
- Split into terms. A term is one part of the expression. The sign in front belongs to the term.5x2 and −3x and +7
- Count them: (a). There are 3 terms.
- Find the coefficients: (b) and (c). The coefficient is the number multiplying the letters, including its sign.In 5x2 the coefficient is 5. In −3x the coefficient is −3 (the minus sign belongs to it).
- Find the constant term: (d). The constant is the term with no letter.The constant term is +7.
- Say what kind of thing it is: (e). There is no equals sign, so it cannot be an equation. A formula needs an equals sign too.It is an expression.
Check. Join the terms: 5x2 + (−3x) + 7 = 5x2 − 3x + 7. ✓
Watch out. The coefficient of −3x is −3, not 3. And the coefficient of a lone x is 1.
Key words. An expression has no equals sign. An equation is true for particular values. A formula is a rule linking quantities. An identity is true for every value.
Method. Look for the sign in the middle: none, =, ≡ or an inequality sign, and think about what the statement does.
- (a) 4a − 9. There is no equals sign and no inequality sign. It is just a collection of terms.Expression
- (b) 2x + 1 = 15. It has an equals sign and is only true for one value of x (x = 7). Something you can solve is an equation.Equation
- (c) P = 2l + 2w. It links several quantities (perimeter, length and width) and you can use it by substituting numbers. A rule like this is a formula.Formula
- (d) 3(x + 2) ≡ 3x + 6. The sign ≡ means "is always equal to". Expanding the bracket gives 3x + 6, so both sides are the same for every x.Identity
- (e) y − 2 < 7. The sign < compares the two sides. It is true for lots of values of y.Inequality
Check. Test the equation in (b): x = 7 gives 2 × 7 + 1 = 15, but x = 3 gives 7, which is not 15. So it is only true for some values. ✓
Watch out. A formula and an equation both have an equals sign. The difference is the job: an equation is solved to find the unknown, and a formula is a general rule you substitute into.
Method. Break the problem into the fixed part and the part that changes with the number of miles.
- Find the part that does not change. The fixed fee is paid whatever the distance.fixed part = 3
- Find the part that changes. Each mile costs £2, so m miles cost 2 × m.changing part = 2 × m = 2m
- (a) Add them. The total is the fixed part plus the changing part.Expression: 3 + 2m (or 2m + 3)
- (b) Make it a formula. A formula has an equals sign and says what the letter C stands for.C = 3 + 2m
- (c) Change pence to pounds first. 80p = £0.80 = £45, and we write the coefficient as a fraction or keep to pence. Here: 80p × m = 80m pence, which is £0.8m.Expression: 0.8m pounds (this is a price, so a decimal is sensible)
Check. Try m = 4 miles: the first taxi costs 3 + 2 × 4 = £11. Counting it out: £3 + four lots of £2 = 3 + 8 = £11. ✓
Watch out. Keep the units the same. Do not mix pounds and pence in one expression.
Method. Use place value: the tens digit is worth ten times its face value.
- Think of a real example. In 53 the tens digit is 5 and the units digit is 3. The value is 5 × 10 + 3.53 = 5 × 10 + 3
- (a) Do the same with letters. Tens digit t is worth 10t. The units digit u is worth u.10t + u
- (b) Swap the digits. Now u is in the tens place and t is in the units place.10u + t
- (c) Add the two expressions. Collect the like terms: the t terms together and the u terms together.(10t + u) + (10u + t) = 10t + t + 10u + u = 11t + 11u
Check. Try 53 and 35: 53 + 35 = 88. The expression gives 11 × 5 + 11 × 3 = 55 + 33 = 88. ✓
Watch out. 10t means 10 × t. Do not write t10.
Watch out
3a and a3 are not the same. 3a means a + a + a and a3 means a × a × a. Try a = 2: 3a = 6 but a3 = 8. Also, 4a means 4 × a, not 4 + a.At Higher tier you are also expected to write expressions for consecutive numbers, even numbers and odd numbers. If n is a whole number, then n + 1 is the next whole number, 2n is always even (it is 2 lots of something) and 2n + 1 is always odd. You will use these in the proofs at the end of this chapter.
Exercise 7A Notation and vocabulary
Non-calculatorMethod. Use the short-cut that matches: leave out the × sign, write the number first, and write repeated letters as a power.
- Spot the type of short-cut. A product of two letters: just join the letters together.
- Write it. Apply the rule.ab
Check. Put a = 2, b = 5: 2 × 5 = 10 and ab = 2 × 5 = 10. ✓
Watch out. Number first, then the letters.
Method. Use the short-cut that matches: leave out the × sign, write the number first, and write repeated letters as a power.
- Spot the type of short-cut. A number times a letter: write the number first, with no × sign.
- Write it. Apply the rule.3c
Check. Put c = 4: 3 × 4 = 12 and 3c = 12. ✓
Watch out. Number first, then the letters.
Method. Use the short-cut that matches: leave out the × sign, write the number first, and write repeated letters as a power.
- Spot the type of short-cut. Four copies of d added: that is 4 lots of d.
- Write it. Apply the rule.4d
Check. Put d = 5: 5 + 5 + 5 + 5 = 20 and 4d = 20. ✓
Watch out. Number first, then the letters.
Method. Use the short-cut that matches: leave out the × sign, write the number first, and write repeated letters as a power.
- Spot the type of short-cut. Two copies of p multiplied: that is p squared.
- Write it. Apply the rule.p2
Check. Put p = 6: 6 × 6 = 36 and 62 = 36. ✓
Watch out. Number first, then the letters.
Method. Use the short-cut that matches: leave out the × sign, write the number first, and write repeated letters as a power.
- Spot the type of short-cut. A number and two letters multiplied: number first, then the letters.
- Write it. Apply the rule.4mn
Check. Put m = 2, n = 3: 4 × 2 × 3 = 24 and 4mn = 24. ✓
Watch out. Number first, then the letters.
Method. Use the short-cut that matches: leave out the × sign, write the number first, and write repeated letters as a power.
- Spot the type of short-cut. A division is written as a fraction.
- Write it. Apply the rule.w7
Check. Put w = 14: 14 ÷ 7 = 2, and the fraction 14 over 7 is also 2. ✓
Watch out. Number first, then the letters.
Method. The index tells you how many copies of the letter are multiplied.
- Read the index. The index is 3, so there are three copies.a3
- Write the copies with × signs. a × a × a
Check. With a = 2: 23 = 8 and 2 × 2 × 2 = 8. ✓
Method. The index only belongs to the letter it sits on, so the 5 is not squared.
- Split into the number and the letter part. 5b2 is 5 multiplied by b2.5 × b2
- Write b2 in full. Two copies of b.5 × b × b
Check. With b = 3: 5 × 32 = 45 and 5 × 3 × 3 = 45. ✓
Watch out. 5b2 is not (5b)2.
Method. Write each letter as many times as its index says.
- Read the indices. x has index 2 and y has index 1 (no index means 1).x2y
- Write the copies. x × x × y
Check. With x = 2, y = 3: 4 × 3 = 12 and 2 × 2 × 3 = 12. ✓
Method. A number next to a bracket means multiply the whole bracket by that number.
- Find the multiplication. There is no sign between 3 and the bracket, so it is a product.3(a + b)
- Write the sign in. 3 × (a + b)
Check. With a = 1, b = 2: 3 × 3 = 9 either way. ✓
Method. A fraction line means divide the top by the bottom.
- Identify the top and the bottom. The top is p and the bottom is q.
- Write as a division. p ÷ q
Check. With p = 12, q = 4: 12 over 4 is 3 and 12 ÷ 4 is 3. ✓
Method. Turn the decimal into a fraction, then multiply.
- Write the decimal as a fraction. 0.5 is five tenths, which simplifies to a half.0.5 = 510 = 12
- Multiply by x. A half of x is written x over 2.12x = x2
Check. With x = 8: 0.5 × 8 = 4 and 8 over 2 is 4. ✓
Method. Turn the decimal into a fraction, then multiply.
- Write the decimal as a fraction. 0.25 is twenty-five hundredths, which simplifies to a quarter.0.25 = 25100 = 14
- Multiply by y. 14y = y4
Check. With y = 8: 0.25 × 8 = 2 and 8 over 4 is 2. ✓
Method. Turn the decimal into a fraction, then multiply.
- Write the decimal as an improper fraction. 1.5 is one and a half, which is 3 halves.1.5 = 32
- Multiply by a. The 3 goes with the a on the top.32a = 3a2
Check. With a = 4: 1.5 × 4 = 6 and (3 × 4) over 2 = 6. ✓
Method. Turn the decimal into a fraction, then multiply.
- Write the decimal as a fraction. 0.2 is two tenths, which simplifies to one fifth.0.2 = 210 = 15
- Multiply by m. 15m = m5
Check. With m = 10: 0.2 × 10 = 2 and 10 over 5 is 2. ✓
Method. Count the parts separated by + or −.
- Split at each + or −. Each part, with its sign, is a term.7x2, +3x, −9
- Count. 3 terms
Check. Three terms, so there are two signs between them, and the expression has two signs: + and −. ✓
Method. The coefficient is the number multiplying the letters.
- Find the x2 term. 7x2
- Read the number in front. coefficient = 7
Check. 7x2 means 7 × x2. ✓
Method. The coefficient is the number multiplying the letters.
- Find the x term. +3x
- Read the number in front. coefficient = 3
Check. 3x means 3 × x. ✓
Method. The constant is the term with no letter. Keep its sign.
- Find the term with no letter. −9
- Keep the sign. The sign belongs to the term.constant term = −9
Check. Without a letter, its value never changes. ✓
Watch out. The constant term is −9, not 9.
Method. Look for an equals sign.
- Look for an equals sign. There is none.
- Decide. Without an equals sign it is not an equation or a formula.It is an expression.
Check. If it said "= 0" at the end it would be an equation. ✓
Key words. Expression: no equals sign. Equation: true for particular values. Formula: a rule linking quantities. Identity: true for all values (≡). Inequality: uses < > ≤ ≥.
Method. Look at the sign in the middle (or the lack of one) and think about what the statement is used for.
- Look at the statement. 5x + 2
- Decide. It has no equals sign or inequality sign.It is an expression.
Check. The five words are: expression, equation, formula, identity, inequality. Only one of them fits. ✓
Key words. Expression: no equals sign. Equation: true for particular values. Formula: a rule linking quantities. Identity: true for all values (≡). Inequality: uses < > ≤ ≥.
Method. Look at the sign in the middle (or the lack of one) and think about what the statement is used for.
- Look at the statement. 3x + 2 = 11
- Decide. It has an equals sign and is true only when x = 3.It is an equation.
Check. The five words are: expression, equation, formula, identity, inequality. Only one of them fits. ✓
Key words. Expression: no equals sign. Equation: true for particular values. Formula: a rule linking quantities. Identity: true for all values (≡). Inequality: uses < > ≤ ≥.
Method. Look at the sign in the middle (or the lack of one) and think about what the statement is used for.
- Look at the statement. P = 2l + 2w
- Decide. It is a rule linking the perimeter P to the length l and width w.It is a formula.
Check. The five words are: expression, equation, formula, identity, inequality. Only one of them fits. ✓
Key words. Expression: no equals sign. Equation: true for particular values. Formula: a rule linking quantities. Identity: true for all values (≡). Inequality: uses < > ≤ ≥.
Method. Look at the sign in the middle (or the lack of one) and think about what the statement is used for.
- Look at the statement. 2(x + 3) ≡ 2x + 6
- Decide. The sign ≡ says both sides are equal for every value of x.It is an identity.
Check. The five words are: expression, equation, formula, identity, inequality. Only one of them fits. ✓
Key words. Expression: no equals sign. Equation: true for particular values. Formula: a rule linking quantities. Identity: true for all values (≡). Inequality: uses < > ≤ ≥.
Method. Look at the sign in the middle (or the lack of one) and think about what the statement is used for.
- Look at the statement. x > 4
- Decide. The sign > compares the two sides.It is an inequality.
Check. The five words are: expression, equation, formula, identity, inequality. Only one of them fits. ✓
Key words. Expression: no equals sign. Equation: true for particular values. Formula: a rule linking quantities. Identity: true for all values (≡). Inequality: uses < > ≤ ≥.
Method. Look at the sign in the middle (or the lack of one) and think about what the statement is used for.
- Look at the statement. 7a − 3b
- Decide. It has no equals sign or inequality sign.It is an expression.
Check. The five words are: expression, equation, formula, identity, inequality. Only one of them fits. ✓
Method. "More than" means add.
- Pick the operation. "More than" means the amount is increased, so we add.x + 7
- Check the order. Addition can go either way round, so 7 + x is fine too.x + 7
Check. If x = 10 then 7 more than 10 is 17, and 10 + 7 = 17. ✓
Method. "Less than" means subtract, and the number after "than" is where you start.
- Find the starting amount. 5 less than y starts from y.y
- Take away 5. y − 5
Check. If y = 12 then 5 less is 7, and 12 − 5 = 7. ✓
Watch out. "5 less than y" is y − 5, not 5 − y.
Method. "Times" means multiply, and we write the number first.
- Pick the operation. 4 × n
- Write it in short form. 4n
Check. If n = 3: 4 times 3 = 12 and 4n = 12. ✓
Method. Divided by means a fraction, with the dividend on top.
- Pick the operation. m ÷ 3
- Write as a fraction. m3
Check. If m = 15 then 15 ÷ 3 = 5 and 15 over 3 is 5. ✓
Method. Do the steps in the order the words give them.
- Double p. Double means multiply by 2.2p
- Add 1. 2p + 1
Check. If p = 4: double is 8, add 1 gives 9, and 2 × 4 + 1 = 9. ✓
Method. "Product" means multiply.
- Find the product. The product of a and b is a × b = ab.ab
- Take 3 away. ab − 3
Check. If a = 2, b = 5: product 10, then 3 less is 7, and 2 × 5 − 3 = 7. ✓
Method. Cost of several items = number of items × cost of one.
- Identify the cost of one and the number bought. One pencil costs p pence; we buy 6.6 × p
- Write in short form. 6p pence
Check. If a pencil costs 20p: 6 × 20 = 120p, and 6p = 6 × 20 = 120. ✓
Method. Add the cost of the ruler to the cost of the pencils.
- Cost of the pencils. From part (a).6p
- Add the ruler. The ruler is a plain number of pence.6p + 40 pence
Check. If p = 20: 120 + 40 = 160p, and 6 × 20 + 40 = 160. ✓
Method. Change = amount paid − amount spent.
- Amount paid. 500
- Amount spent. From part (a).6p
- Subtract. Write 500 first because we start with 500 and take away.500 − 6p pence
Check. If p = 20: 500 − 120 = 380, and 500 − 6 × 20 = 380. ✓
Watch out. 500 − 6p is not the same as 6p − 500.
Method. A formula has an equals sign: write what the letter equals.
- Fixed part. The fee that is paid whatever the distance.3
- Changing part. £2 for each of the m miles.2 × m = 2m
- Put together with an equals sign. C = 3 + 2m
Check. A 1 mile journey: 3 + 2 = £5, which is the fee plus one mile. ✓
Method. Substitute m = 5 into the formula.
- Replace m by 5. Substituting means putting a number in place of a letter.C = 3 + 2 × 5
- Multiply first (BIDMAS). 2 × 5 = 10
- Add. 3 + 10 = 13
Check. Fee £3 plus 5 miles at £2 each (£10) is £13. ✓
Method. Perimeter is the distance all the way round. Add the four sides.
- Add the four sides. Two sides have length l and two have width w.l + w + l + w
- Collect the same letters. l + l = 2l and w + w = 2w.P = 2l + 2w
Check. A 5 by 3 rectangle: 5 + 3 + 5 + 3 = 16 and 2 × 5 + 2 × 3 = 16. ✓
Method. Area of a rectangle = length × width.
- Write length times width. l × w
- Short form. Leave out the × sign.A = lw
Check. A 5 by 3 rectangle: 5 × 3 = 15 square units. ✓
Method. "5 years older" means add 5.
- Start with Amy's age. a
- Add 5. a + 5
Check. If Amy is 10, Ben is 15, and 10 + 5 = 15. ✓
Method. Twice Ben's age means 2 times the whole of Ben's age, so we need a bracket.
- Ben's age. From part (a).a + 5
- Multiply by 2. The 2 must multiply the whole of a + 5, so we put it in brackets.2(a + 5)
Check. If Amy is 10: Ben is 15, Cara is 30, and 2(10 + 5) = 30. ✓
Watch out. 2a + 5 would mean double Amy's age, then add 5. That is a different thing.
Method. Substitute a number into both expressions. If they give different answers they are not the same.
- Work out 4a with a = 3. 4a means 4 × a.4 × 3 = 12
- Work out 4 + a with a = 3. 4 + 3 = 7
- Compare. 12 is not equal to 7.So 4a is not the same as 4 + a.
- Say what 4a means. A number next to a letter, with no sign, means multiply.4a means 4 × a, which is also a + a + a + a.
Check. a + a + a + a = 3 + 3 + 3 + 3 = 12, which matches 4a. ✓
Watch out. One example that fails is enough to show that a statement is false.
Method. Test with numbers, then think about why.
- Test a = 3. 3 + 3 = 6 and 2 × 3 = 6
- Explain. Two copies of a added together is 2 lots of a. This works for any a.True
Check. Test a = 10: 20 and 20. ✓
Method. Test with numbers. One value that fails shows it is false.
- Test a = 3. 3 × 3 = 9 but 2 × 3 = 6
- Conclude. 9 is not 6, so the two sides are not always equal.False. a × a = a2.
Check. It does work for a = 2 (4 and 4), but a statement must work for all values to be true. ✓
Method. Test with numbers.
- Test a = 2. 3 × 2 = 6 but 23 = 2 × 2 × 2 = 8
- Conclude. 6 is not 8.False
Check. 3a means a + a + a; a3 means a × a × a. ✓
Method. Test with numbers.
- Test a = 6 and b = 3. 6 ÷ 3 = 2 but 3 ÷ 6 = 12
- Conclude. They are different.False
Check. Division is not the same both ways round. ✓
Key words. An equation is true for particular values only. An identity is true for all values.
Method. An identity is true for every value. If one value fails, it is not an identity.
- Try x = 4. Replace x by 4 on the left side.3 × 4 + 1 = 12 + 1 = 13, which equals the right side. ✓
- Try x = 5. 3 × 5 + 1 = 15 + 1 = 16, which is not 13. ✗
- Conclude. It is true for x = 4 but not for x = 5, so it is not true for every value.It is an equation (true for a particular value), not an identity.
Check. An identity would have to work for every number, for example 3(x + 2) = 3x + 6 does. ✓
Method. The next whole number is 1 more.
- Think of a number example. After 7 comes 8, which is 7 + 1.
- Do the same with the letter. n + 1
Check. If n = 7 the next is 8 = 7 + 1. ✓
Method. An even number is 2 times a whole number.
- Look at the pattern. 2, 4, 6, 8 are 2 × 1, 2 × 2, 2 × 3, 2 × 4.
- Write it with a letter. 2n
Check. If n = 5: 2n = 10, which is even. ✓
Method. An odd number is one more than an even number.
- Start with an even number. 2n
- Add 1. Adding 1 to an even number gives an odd number.2n + 1
Check. If n = 5: 2n + 1 = 11, which is odd. ✓
Method. Consecutive even numbers differ by 2.
- Start with the even number. 2n
- Add 2. 2n + 2
Check. If n = 5: 10 then 12 = 10 + 2. ✓
Watch out. 2n + 2 can also be written 2(n + 1).
Method. Consecutive whole numbers go up by 1 each time.
- Start with the smallest. n
- Add 1 each time. n, n + 1, n + 2
Check. If n = 7: 7, 8, 9. ✓
Method. Use place value to write each number, then collect like terms.
- The number. The tens digit is worth 10 times as much.10t + u
- The swapped number. 10u + t
- Add them. Collect the t terms and the u terms.10t + u + 10u + t = 11t + 11u
Check. 62 + 26 = 88 and 11 × 6 + 11 × 2 = 66 + 22 = 88. ✓
Method. Show that the two forms give the same number, and say why: 2 times the whole bracket is 2 times each part.
- Work out 2(l + w). Brackets first.2(7 + 4) = 2 × 11 = 22
- Work out 2l + 2w. 2 × 7 + 2 × 4 = 14 + 8 = 22
- Explain. The 2 multiplies the whole bracket, so it multiplies both the l and the w. This is what "expanding" means.Both forms give 22, and 2(l + w) = 2l + 2w.
Check. Going round a 7 by 4 rectangle: 7 + 4 + 7 + 4 = 22. ✓